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Well-posedness and decay of solutions to 3D generalized Navier-Stokes equations

  • * Corresponding author: Xiaopeng Zhao

    * Corresponding author: Xiaopeng Zhao 

This paper is supported by the National Nature Science Foundation of China (grant No. 11401258), Nature Science Foundation of Jiangsu Province (grant No. BK20140130) and China Postdoctoral Science Foundation (grant No. 2015M581689)

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  • The global well-posedness and large time behavior of solutions for the Cauchy problem of the three-dimensional generalized Navier-Stokes equations are studied. We first construct a local continuous solution, then by combining some a priori estimates and the continuity argument, the local continuous solution is extended to all $ t>0 $ step by step provided that the initial data is sufficiently small. In addition, by using Strauss's inequality, generalized interpolation type lemma and a bootstrap argument, we establish the $ L^p $ decay estimate for the solution $ u(\cdot,t) $ and all its derivatives for generalized Navier-Stokes equations with $ \max\{1,\frac{3+q}6\}<\alpha\leq\frac12+\min\{\frac3q-\frac3p,\frac3{2p}\} $.

    Mathematics Subject Classification: Primary: 35B65; Secondary: 35Q35.


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